Noncommutative Bennett and Rosenthal inequalities

TL;DR

This paper extends classical inequalities to the noncommutative setting, improves constants in Rosenthal's inequality, and applies these results to random Fourier projections in compressed sensing.

Contribution

It introduces noncommutative versions of Bennett, Bernstein, and Prohorov inequalities and refines the constants in the noncommutative Rosenthal inequality.

Findings

Extended classical inequalities to noncommutative variables

Provided improved constants in noncommutative Rosenthal inequality

Applied results to random Fourier projections in compressed sensing

Abstract

In this paper we extend the Bernstein, Prohorov and Bennett inequalities to the noncommutative setting. In addition we provide an improved version of the noncommutative Rosenthal inequality, essentially due to Nagaev, Pinelis and Pinelis, Utev for commutative random variables. We also present new best constants in Rosenthal's inequality. Applying these results to random Fourier projections, we recover and elaborate on fundamental results from compressed sensing, due to Candes, Romberg and Tao.